Mon. Not. R. Astron. Soc. 375, 1315–1324 (2007) doi:10.1111/j.1365-2966.2006.11388.x
White dwarf mass distribution in the SDSS
S. O. Kepler,1� S. J. Kleinman,2 A. Nitta,3 D. Koester,4 B. G. Castanheira,1
O. Giovannini,5 A. F. M. Costa1 and L. Althaus6
1Instituto de Fısica, Universidade Federal do Rio Grande do Sul, 91501-900 Porto-Alegre, RS, Brazil2Subaru Telescope, 650 N. A’Ohoku Place, Hilo, HI 96720, USA3Gemini Observatory, Hilo, HI 96720, USA4Institut fur Theoretische Physik und Astrophysik, Universitat Kiel, 24098 Kiel, Germany5Universidade de Caxias do Sul, 95070-560 Caxias do Sul, RS, Brazil6Facultad de Ciencias Astronomicas y Geofısicas, Paseo del Bosque S/N, 1900 La Plata, Argentina
Accepted 2006 December 6. Received 2006 November 22; in original form 2006 August 1
ABSTRACTWe determined masses for the 7167 DA and 507 DB white dwarf stars classified as single and
non-magnetic in Data Release 4 of the Sloan Digital Sky Survey (SDSS). We obtained revised
Teff and log g determinations for the most massive stars by fitting the SDSS optical spectra
with a synthetic spectra grid derived from model atmospheres extending to log g = 10.0. We
also calculate radii from evolutionary models and create volume-corrected mass distributions
for our DA and DB samples. The mean mass for the DA stars brighter than g = 19 and hotter
than Teff = 12 000 K is 〈M〉DA � 0.593 ± 0.016M�. For the 150 DBs brighter than g = 19
and hotter than Teff = 16 000 K, we find 〈M〉DB = 0.711 ± 0.009M�. It appears the mean
mass for DB white dwarf stars may be significantly larger than that for DAs. We also report
the highest mass white dwarf stars ever found, up to 1.33M�.
Key words: white dwarfs.
1 I N T RO D U C T I O N
White dwarf stars are the end product of evolution of all stars with
initial masses up to around 9M� and their distribution contains
information about star formation history and subsequent evolution
in our Galaxy. As the most common endpoints of stellar evolution,
white dwarf stars account for around 97 per cent of all evolved stars.
Considering there has not yet been enough time for any of them to
cool down to undetectability, they can also provide independent
information about the age of the Galaxy. Through an initial–final
mass relation (IFMR), we can also study mass loss throughout the
stellar evolution process. Because white dwarf progenitors lose car-
bon, nitrogen and oxygen at the top of the asymptotic giant branch
(AGB), they are significant contributors to the chemical evolution
of the Galaxy and possibly an important source of life sustaining
chemicals.
Kleinman et al. (2004) published the spectra of 2551 white dwarf
stars in the Sloan Digital Sky Survey (SDSS) Data Release 1 (DR1),
covering 1360 deg2. Eisenstein et al. (2006) extended the white
dwarf spectroscopic identifications to the SDSS Data Release 4
(DR4) with a total of 9316 white dwarf stars reported, more than
doubling the number of spectroscopically identified stars (McCook
& Sion 2003). In both works, the authors fit the entire optical spectra
from 3900 to 6800 Å to DA and DB grids of synthetic spectra de-
�E-mail: [email protected]
rived from model atmospheres calculated by Detlev Koester, up to
log g = 9.0 (in cgs units). Their fits include SDSS imaging photome-
try and allow for refluxing of the models by a low-order polynomial
to incorporate effects of unknown reddening and spectrophotomet-
ric errors. The SDSS spectra have a mean g-band signal-to-noise
ratio SNR (g) ≈ 13 for all DAs, and SNR (g) ≈ 21 for those brighter
than g = 19.
This large sample of stars with spectroscopic fits gives us a new
opportunity to fully explore the white dwarf mass distribution. Un-
derstanding the white dwarf mass distribution offers insights into
mass loss during stellar evolution, the IFMR and has bearings on
close binary star evolution. Our report, as well as many previous
studies, detect a substantial fraction of low-mass white dwarf stars
that theoretically cannot have evolved as single stars, because the
age of the Universe is smaller than their presumed lifetimes on the
main sequence.
Kleinman et al. (2004) notice an increase in mean log g for stars
cooler than Teff = 12 000 K, but caution the trend might not be real,
indicating a problem in the data or fit technique, instead. The trend
has persisted into the larger catalogue of Eisenstein et al. (2006).
Madej, Nalezyty & Althaus (2004) analysed the Kleinman et al.
(2004) sample of fits and calculated the corresponding SDSS DR1
pure hydrogen atmosphere (DA) white dwarf mass distribution. As
expected from the log g trend, they found that the mean mass also
increased below Teff = 12 000 K. Their table 1 presents all previous
mean mass determinations, producing an average of 0.57M�, and
a most populated peak at 0.562M� for the 1175 stars hotter than
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1316 S. O. Kepler et al.
Teff = 12 000 K. They did not study the potentially highest mass stars
with log g > 9, because they were limited by the stellar atmosphere
fit by Kleinman et al. (2004) which artificially pegged stars near the
upper log g = 9.0 boundary to the boundary itself.
The increase in mean masses fitted from optical spectra below
Teff = 12 000 K has been seen prior to Kleinman et al. (2004) and
has been discussed since Bergeron, Wesemael & Fontaine (1991)
and Koester (1991). It is usually dismissed as due to problems in the
models: either convection bringing up subsurface He to the atmo-
sphere, increasing the local pressure, or problems with the treatment
of the hydrogen level occupation probability. The new larger SDSS
data set, however, now gives another opportunity to explore this
trend and evaluate its cause.
Most reported white dwarf mass determinations have been de-
rived by comparing the optical spectra with model atmospheres, as
with Kleinman et al. (2004) and Eisenstein et al. (2006). For the
DA stars, the H7, H8 and H9 lines, in the violet, are the most sen-
sitive to surface gravity because they are produced by electrons at
higher energy levels, those most affected by neighbouring atoms.
However, these lines are also in the region where the atmospheric
extinction is the largest and typical CCD detectors are the least sen-
sitive. As a consequence, most studies used only the line profiles
in their fits, avoiding the dependence on often uncertain flux cali-
brations. The SDSS white dwarf spectra have good flux calibration
and acceptable SNR redwards of 4000 Å. The published SDSS cat-
alogue therefore fits the entire optical spectrum, and not just the H
lines, as has been traditionally done. The rationale for this approach
is the good, uniform spectrophotometry and corresponding broad-
band photometry that can be used in the fits. In addition, a low-level
refluxing is allowed to take out large errors in spectrophotometry
and any unknown reddening effects.
In this paper, we will compare the measured white dwarf mass
distributions from Kleinman et al. (2004) and Eisenstein et al. (2006)
with previous determinations and attempt to assess the reason for
the observed increase in mass for lower temperatures. We will also
explore the observed mean masses and analyse the two different
fitting techniques: line profile versus whole spectrum, to see the
effects on the resulting mass distributions.
2 DATA A N D M O D E L S
The SDSS imaged the high Galactic latitude sky in five pass bands: u,g, r, i and z, and obtained spectra from 3800 to 9200 Å with a resolu-
tion of ≈1800 using a twin fibre-fed spectrograph (York et al. 2000).
Since we are primarily interested in the mass distribution here, we
selected only the single DA and DB stars with log g − σ logg � 8.5
and log g + σ logg � 6.5 from the Eisenstein et al. (2006) sample and
refit them with an expanded grid of models (see below), using the
same AUTOFIT routine as in Eisenstein et al. (2006) and thoroughly
described in Kleinman et al. (2004). We excluded all stars classified
by Eisenstein et al. (2006) as having either a detectable magnetic
field or a companion, metal lines, DABs and DBAs.
Our model grid (Finley, Koester & Basri 1997; Koester et al. 2001)
is similar to that used by Eisenstein et al. (2006), but extended in
Teff and log g (100 000 K � Teff � 6000 K, 10.0 � log g � 5.0) and
denser. We chose the ML2/α = 0.6 parametrization for convection as
demonstrated by Bergeron et al. (1995) to give internal consistency
between temperatures derived in the optical and the ultraviolet, pho-
tometry, parallax and with gravitational redshift. ML2 corresponds
to the Bohm & Cassinelli (1971) description of the mixing length
theory and α = �/λP is the ratio of the mixing length to the pressure
scaleheight. The models include the H+2 and H2 quasi-molecular
opacities and only Stark (Lemke 1997) and Doppler broadening, so
the line profiles are not precise for Teff < 8500 K.
Even though Napiwotzki, Green & Saffer (1999) and Liebert,
Bergeron & Holberg (2005) discuss the necessity of using non-local
thermodynamic equilibrium (NLTE) atmospheres for the stars hot-
ter than 40 000 K, all quoted values are from local thermodynamic
equilibrium (LTE) models, as they also show the NLTE corrections
are not dominant, and our number of hot stars is small.
To calculate the mass of each star from the Teff and log g values
obtained from our fits, we used the evolutionary models of Wood
(1995) and Althaus et al. (2005) with C/O cores up to log g = 9.0, and
O/Ne cores for higher gravities, MHe = 10−2 M∗ and MH = 10−4 M∗or MH = 0, to estimate stellar radii for DAs and DBs, respectively.
The radius is necessary to convert surface gravity to mass.
3 A NA LY S I S
Before exploring the mass distributions, we wanted to examine the
different fitting techniques used in the available data sets – the tradi-
tional line profile technique and the SDSS whole spectrum approach.
We therefore simulated spectra with differing SNRs by adding ran-
dom noise to our models and fit them with our own set of both line
profile and whole spectrum fitting routines. Our Monte Carlo sim-
ulations show that in the low SNR regime, SNR � 50, fitting the
whole spectra and not just the line profiles gives more accurate at-
mospheric parameters, as long as the flux calibration or interstellar
reddening uncertainties do not dominate. We estimate an uncertainty
of around �Teff � 500 K and � log g � 0.10 at SNR = 40 for the
whole spectra fitting. For SNR = 20, similar to the average SDSS
spectra for g < 19, our simulations indicate �Teff � 750 K and
� log g � 0.15. We do not report in this paper on the mass distribu-
tion for the stars fainter than g = 19 because their smaller SNR lead
to large uncertainties. Our simulations did not indicate systematic
trends between the two approaches.
Although Kleinman et al. (2004) and Eisenstein et al. (2006)
compared their fits’ internal errors by fitting duplicate spectra, they
did not display their results as a function of temperature. Kepler
et al. (2005) specifically analysed 109 duplicate spectra SDSS DAs
with 13 000 � Teff � 10 000 K, near the region where the fit log gs
start to increase. They showed that the mean fit differences were
σTeff� 300 K and σ logg � 0.21 dex for the same object but different
observations. These values are larger than the internal uncertainty
of the fits, but in general within 3σ of each other, as in Kleinman
et al. (2004) and Eisenstein et al. (2006). We thus conclude that the
uncertainties in Eisenstein et al. (2006) are reasonable and can now
analyse the results without attributing any noted irregularities to the
fitting process.
Kepler et al. (2006), however, compare SDSS spectra with new
SNR (g) � 100 spectra acquired with Gemini Multi-Object Spec-
trograph (GMOS) on the Gemini 8-m telescope for four white dwarf
stars around Teff � 12 000 K. Their fits suggest that published SDSS
optical spectra fits overestimate the mass by �M � 0.13 M�, be-
cause of the correlation between the derived Teff and log g – a small
increase in Teff can be compensated by a small decrease in log g. In
our simulations this discrepancy is concentrated only in the region
around the Balmer line maximum, 14 000 � Teff � 11 000 K.
To explore the increasing mass trend in more detail, we restricted
our sample to the 1733 stars both brighter than g = 19 and hot-
ter than Teff = 12 000 K and obtained an average DA mass of
〈M〉DA = 0.593 ± 0.016M�. The distribution for this hot and
bright sample, shown in Fig. 1, is similar to that of the Palomar
Green survey published by Liebert et al. (2005). They studied a
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White dwarf mass distribution in the SDSS 1317
0.2 0.4 0.6 0.8 1 1.2 1.4
0
100
200
300
SDSS DR4
DR1
PG
0.2 0.4 0.6 0.8 1 1.2 1.4
0
100
200
300
Figure 1. Histogram for the 1859 DA stars brighter than g = 19 and hotter
than Teff = 12 000 K, compared to the PG survey published by Liebert
et al. (2005) and the SDSS DR1 sample published by Madej et al. (2004).
Gaussian fits detailed in Table 1 are also shown. Our bins are 0.025-M�wide. The second graph shows the DR1 and PG survey data normalized to
the DR4 sample, even though those samples are smaller and therefore have
significantly larger error bars.
complete sample of 348 DA stars with SNR � 60 spectra and de-
termined atmospheric parameters by spectral fitting via the line
profile fitting technique, using models up to log g = 9.5. They
found a peak in the mass histogram at 0.565M� containing 75 per
cent of the sample, a low-mass peak with 0.403M� containing
40000 30000 20000 10000
0.5
1.0
DR4 DAs and PG Survey
DA
Figure 2. Masses for all 3595 DA white dwarf stars brighter than g = 19 and
cooler than 40 000 K, showing an increase in mean mass for lower Teff. The
large solid (blue) circles are the values published by Liebert et al. (2005),
showing the increase in mass at lower Teff is also present in their sample,
which uses a totally independent grid of models and fitting technique.
10 per cent of the sample, and a high-mass peak at 0.780M�containing 15 per cent of the stars. They fit their mass histogram
(PG mass histogram from hereafter) with three Gaussian pro-
files: 0.565M� with σ � 0.080M�, 0.403M� with σ �0.023M� and a broad high-mass component at 0.780M� with
σ � 0.108M�. They found more stars above 1M� than can be
described by the three Gaussians they fit. Marsh et al. (1997), Vennes
et al. (1997) and Vennes (1999) also find an excess of white dwarf
stars with masses above 1M� in their sample of Teff > 23 000 K
white dwarf stars.
The overall mass distribution of our bright sample matches well
with that of the previous standard PG survey sample. We now explore
the distribution with temperature.
In Fig. 2, we show the mass distribution versus temperature for DA
stars brighter than g = 19 along with the similar distribution from
Liebert et al. (2005). Again, we see the distributions are roughly
equivalent and we see an increase in measured mass at lower tem-
peratures. Our histograms use 0.025M� bins (N/d m = constant)
because that is the approximately mean uncertainty in our mass
determinations.
To explore the region of increasing mass further, Fig. 3 shows
the mass histogram only for the 964 DAs brighter than g = 19, and
12 000 � Teff � 8500 K, for which we obtain 〈M〉coolDA � 0.789 ±
0.005M�. We have excluded the stars cooler than Teff = 8500 K
from our mass histograms because our cooler atmospheric models
are not accurate for log g determination, as explained earlier.
Tables 1 and 2 detail the Gaussian fits we made for the histograms
of Figs 1 and 3, respectively, with
N =∑
i
ai exp
[− (M − Mi )
2
2σ 2i
]. (1)
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1318 S. O. Kepler et al.
0 0 .5 1
0
20
40
60
80
Figure 3. Histogram for the 964 DA stars brighter than g = 19, with
12 000 � Teff � 8500 K along with the fit Gaussians as detailed in Table 2.
Table 1. Gaussian fits for the Teff � 12 000 K and g � 19
histogram, seen in Fig. 1.
i ai Mi (M�) σi Fraction
1 264.8 0.578 0.047 69 per cent
2 27.8 0.658 0.149 23 per cent
3 27.0 0.381 0.050 7 per cent
4 3.0 1.081 0.143 1 per cent
Table 2. Gaussian fits for the 964 DAs with 12 000 � Teff �8500 K and g � 19 histogram.
i ai Mi (M�) σi Fraction
1 29.5 0.818 0.160 49 per cent
2 59.6 0.793 0.052 33 per cent
3 18.0 0.640 0.086 16 per cent
4 13.4 1.096 0.136 2 per cent
Fig. 11 of Liebert et al. (2005) also shows an increase in mass
below Teff = 12 000 K (see Fig. 2), even though they have a lim-
ited number of cooler stars due to colour selection effects in the PG
survey. It is important to note that the model atmospheres used in
Liebert et al. (2005) and the line profile fitting technique they use are
totally independent of our own. Therefore, if the observed increase
in the measured gravity with temperature is merely an artefact of
the models, then similar problems must be present in two indepen-
dent groups of models and fitting techniques. We are thus gathering
increasing evidence that either (a) both DAs and DBs really do have
higher mean masses at lower temperatures, or (b) there is a common
artefact in the model used for all white dwarf spectral fitting.
4 W H Y W E D O N OT T RU S T M A S S E S F O RT eff < 1 2 0 0 0 K F O R DA S TA R S
Bergeron et al. (1995) measured an increase in the mean mass for
the ZZ Ceti star sample around 13 000 > Teff > 11 000 K, but indi-
cated it might come from a selection effect because the discovery
of pulsating stars might have preferred higher mass stars. Arras,
Townsley & Bildsten (2006) e.g. show that the white dwarf pul-
sators with lower masses should pulsate at cooler temperatures. Our
sample of 351 bright stars in the same temperature range shows a
similar increase in mass compared to the hotter sample, but we have
not been biased by the pulsators, so an observational bias is not the
cause for the increase in mass detected.
The simple expectation that massive stars cool faster than their
less massive counterparts does not hold for Teff � 10 000 K, as the
most massive stars have smaller radius and, therefore, their cooling
slows down after a few e-folding time-scales. Another possible ex-
planation for an increased mean mass at lower effective temperatures
is the presence of otherwise undetected He at the surface, broadening
the observed H lines and thus mimicking a higher log g. Theoreti-
cal models (e.g. Fontaine & Wesemael 1991) indicate that only for
DAs with hydrogen layer masses below MH = 10−10 M∗ will He
mix around Teff = 10 000 K and, if MH = 10−7 M∗, only below
Teff � 6500 K. However, all seismologically measured H layer
masses are MH > 10−7 M∗ (Fontaine & Wesemael 1997; Bradley
1998, 2001, 2006). Since our increased mass trend happens signifi-
cantly hotter than Teff = 6500 K, He contamination cannot account
for the observed increase in mass at lower temperatures, unless the
more distant stars studied here have significantly thinner H layers.
Lawlor & MacDonald’s (2006) models show around 3 per cent of
the DAs could have MH ∼ 10−9 M∗, but not thinner. Therefore,
there are not enough stars with thin H layers, at any rate, to account
for our excess of massive objects.
Wilson (2000) proposes a possible physical model for increasing
white dwarf masses at lower temperatures. She suggests that low-
metallicity AGB stars will produce higher mass white dwarf stars,
probably around 1M�, because the relatively lower mass loss ex-
pected for low metallicity AGB stars increases the mass of the core
prior to the star moving out of the AGB. Since the earlier genera-
tions of white dwarf stars which have now cooled more than their
later cohorts, presumably came from lower metallicity progenitors,
this mechanism could explain a mass increase at lower white dwarf
temperatures. If we extend this concept to globular clusters though,
we would expect the mass of the white dwarfs in globular clusters
to be larger than the mean mass of our stars cooler than 10 000 K,
which is not observed (Moehler et al. 2004; Richer et al. 2006). So
again, we are left with a discarded explanation of the observed mass
increase.
An interesting clue to the problem may be found in Engelbrecht
& Koester (2007), which used SDSS photometry alone to make
a mass estimate. Their cool white dwarf stars show mean masses
similar to those of the hotter stars. Our mass determination using
photometric colours only, shown in Fig. 4, is derived comparing
only the SDSS colours (u − g) and (g − r) with those predicted
from the atmospheric models convolved with SDSS filters. They
do not show any increase in mass with decreasing Teff. Because
of their larger uncertainties than the spectra fitting, we binned the
results in 2000-K bins. This result suggests that any problem in the
models is mainly restricted to the line profiles, not the continua,
which dominate the broad-band photometric colours.
Thus, we are mainly left with the possibility raised by Koester
(1991) that an increase in mass with lower temperatures could be
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White dwarf mass distribution in the SDSS 1319
25000 20000 15000 100000.4
0.6
0.8
1.0
1.2
SDSS DR4 DAs Masses from Colors
Figure 4. Masses for all 7167 DA white dwarf stars derived comparing
only the SDSS colours (u − g) and (g − r) with those predicted from the
atmospheric models. For Teff � 20 000 K and Teff � 9000 K, the colours are
degenerate in mass.
due to the treatment of neutral particles in model atmospheres with
the Hummer–Mihalas formalism. Bergeron et al. (1995), however,
suggest that the neutral particles are only important below Teff �8000 K which is certainly lower than where we see the trend begin. It
seems the only remaining explanation is that accurate modelling of
neutral particles will indeed show an effect for DAs near 12 000 K.
5 D B W H I T E DWA R F S
We determined masses for the Eisenstein et al. (2006) DBs from
their fit temperatures and gravities using evolutionary grids of
Althaus, Serenelli & Benvenuto (2001) and Althaus et al. (2005).
The Althaus models use time resolved diffusion throughout evolu-
tion. Metcalfe (2005) and Metcalfe et al. (2005) discuss asteroseis-
mological results in DBs, showing the observations are consistent
with the layer masses predicted by current diffusion theory. Fig. 5
shows that we find an increase in the measured surface grav-
ity below Teff � 12 000 K for DAs and a similar increase below
Teff � 16 000 K for DBs. For the 208 DBs brighter than g =19, we find 〈M〉all
DB = 0.785 ± 0.013M�. For the 150 DBs
brighter than g = 19 and hotter than Teff = 16 000 K, we find
〈M〉DB = 0.711±0.009M�. Both measurements are considerably
larger than the 0.593M� mean mass value for the bright and hot
DA sample. A similar larger (relative to that of the DAs) DB mean
mass value has been previously reported by Koester et al. (2001) who
obtained a 〈M〉DB = 0.77 for the 18 DBs they observed with the
Ultraviolet and Visual Echelle Spectrograph (UVES)/Very Large
Telescope (VLT), including stars down to Teff ∼ 16 000 K. Others,
however, find lower mean DB masses, more similar to those of the
DAs. Oke, Weidemann & Koester (1984) derived 〈M〉DB = 0.55 ±0.10 from their sample of 25 DBs ranging 30 000 > Teff > 12 000 K,
while Beauchamp (1995) found 〈M〉DB = 0.59 ± 0.01M� for his
40000 30000 20000 10000
0.5
1.0
DR4 DAs and DBs
Figure 5. Masses for all 5718 DAs (crosses) cooler than 40 000 K and 507
DBs (filled circles) showing the continuous increase in average mass at
lower Teff.
Table 3. Gaussian fits for the histogram of the 150 DBs
brighter than g = 19 and hotter than Teff = 16 000 K.
i ai Mi (M�) σi Fraction
1 8.3 0.700 0.109 59 per cent
2 14.3 0.712 0.042 40 per cent
3 0.6 1.288 0.035 1 per cent
46 DBs, ranging 12 000 � Teff � 31 000 K. For the 34 DBs in
Castanheira et al. (2006), ranging 27 000 � Teff � 13 000 K, the
mean is 〈M〉DB = 0.544 ± 0.05M�.
The Gaussian fits for the 153 DBs brighter than g = 19 and hotter
than Teff = 16 000 K are listed in Table 3. The mass histogram for
DBs is shown in Fig. 6.
6 O B S E RV I N G VO L U M E C O R R E C T I O N
In order to turn our observed mass distributions into a real analyt-
ical tool, we must first correct the sample for completeness. We
do this by the 1/Vmax formalism. Vmax is the volume defined by
the maximum distance at which a given object would still appear
in a magnitude limited sample (Schmidt 1968). Geijo et al. (2006)
discuss white dwarf luminosity function completeness corrections
and conclude that for large samples, the 1/Vmax method provides a
reliable characterization of the white dwarf luminosity function.
Liebert et al. (2005) find that 2.7 per cent of the stars in the
PG sample have masses larger than 1M� and, when corrected by
1/Vmax, 22 per cent are above 0.8M�.
We first calculated each star’s absolute magnitude from the Teff
and log g values obtained from our fits (for the extreme mass ones)
or those of Eisenstein et al. (2006) (for the rest), convolving the
synthetic spectra with the g filter transmission curve. We used the
evolutionary models of Wood (1995) and Althaus et al. (2005) with
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1320 S. O. Kepler et al.
0.2 0.4 0.6 0.8 1 1.2 1.4
0
5
10
15
20
25
Figure 6. Histogram for the 150 DBs brighter than g = 19 and hotter than
Teff = 16 000 K in DR4.
C/O cores up to log g = 9.0, and O/Ne cores for higher gravities,
MHe = 10−2 M∗ and MH = 10−4 or 0 M∗, to estimate stellar radii
for DAs and DBs, respectively. We do not claim that the SDSS
spectroscopic sample is complete, but we do contend that in terms
of mass, there should be no preferential bias in the target selection.
Harris et al. (2006) report that spectra are obtained for essentially
all white dwarf stars hotter than 22 000 K. Additional white dwarf
stars down to Teff = 8000 K are also found, but few cooler than that
as these stars overlap in colour space with the F, G and K main-
sequence stars. Eisenstein et al. (2006) discuss the spectroscopic
sample completeness, which is around 60 per cent at 18 < g < 19.5
for stars hotter than Teff = 12 000 K and around 40 per cent for cooler
stars. Our analysis is restricted to the sample brighter than g = 19.
Once we had our calculated absolute magnitudes, we could esti-
mate each star’s distance as shown in Fig. 7, neglecting any effects
of interstellar extinction. The mean distance for our DA samples
are as follows: 474 ± 5 pc for the entire 7167 DA sample; 302 ±5 pc for the stars brighter than g = 19 and 436 ± 7 pc for the stars
brighter than g = 19 and hotter than Teff � 12 000 K.
For each star in our sample, we calculate
Vmax = 4π3
β(r 3
max − r 3min
)exp(−z/z0),
where β is the fraction of the sky covered, 0.1159 for the DR4 sam-
ple, rmin is due to the bright magnitude limit, g = 15, and z0 is the
disc scaleheight which we assume to be 250 pc, as Liebert et al.
(2005) and Harris et al. (2006), even though our height distribution
indicates z0 � 310 pc. Vennes et al. (2005) show that both the white
dwarf stars in the SDSS DR1, and the 1934 DAs found in the 2dF
(18.25 � bJ � 20.85) quasar surveys, belong to the thin disc of
our Galaxy. Using these data, they measured a scaleheight around
300 pc. Harris et al. (2006) calculate the white dwarf luminosity
function from photometric measurements of the white dwarf stars
discovered in the SDSS survey up to Data Release 3 (DR3). They
assume log g = 8.0 for all stars and use the change in number per
0 1000 2000 3000
0
500
1000
1500
2000
d(pc)
Sloan DR4 DAs
Figure 7. Distribution of distances, d, and height above the Galactic plane,
z, for DAs in the SDSS DR4.
0.2 0.4 0.6 0.8 1 1.2 1.4
0
Figure 8. Histogram for the 1733 DA stars brighter than g = 19 (V /Vmax �0.45) and hotter than Teff = 12 000 K corrected by 1/Vmax.
magnitude bin to calculate the scaleheight of the disc, obtaining
340+100−70 pc, but adopt 250 pc for better comparison with other stud-
ies. This volume includes the disc scaleheight as discussed by Green
(1980), Fleming, Liebert & Green (1986) and Liebert et al. (2005).
Each star contributes 1/Vmax to the local space density.
Figs 8 and 9 show the resulting corrected mass distribution for
our DA and DB sample, respectively. Fig. 8 contains 1733 bright,
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White dwarf mass distribution in the SDSS 1321
0.2 0.4 0.6 0.8 1 1.2 1.4
0
SDSS DR4 DB Mass Distribution
Figure 9. Histogram for the 150 DBs brighter than g = 19, and hotter than
Teff = 16 000 K in DR4, corrected by volume.
Table 4. Gaussian fits for the Teff � 12 000 K and g � 19 volume
corrected DA mass histogram.
i ai Mi (M�) σi Fraction
1 5.965 × 10−6 0.603 0.081 38 per cent
2 1.203 × 10−5 0.571 0.034 32 per cent
3 1.165 × 10−6 0.775 0.201 19 per cent
4 1.455 × 10−6 1.175 0.076 9 per cent
5 8.305 × 10−7 0.358 0.037 2 per cent
Table 5. Gaussian fits for the volume corrected histogram of the 150
DBs brighter than g = 19 and hotter than Teff = 16 000 K.
i ai Mi (M�) σi Fraction
1 8.6 × 10−7 0.718 0.111 66 per cent
2 8.9 × 10−7 0.715 0.045 27 per cent
3 2.9 × 10−7 1.286 0.031 7 per cent
non-cool DAs, i.e. those with Teff � 12 000 K and g � 19. We also
list the corresponding five Gaussian fit parameters in Table 4. Fig. 9
contains 150 bright, non-cool DBs, i.e. those with Teff � 16 000 K
and g � 19. The corresponding three Gaussian fits are listed in
Table 5.
Since the most massive white dwarf stars have smaller luminosi-
ties because of their smaller radii, after applying the 1/Vmax cor-
rection to the observed volume, we find that around 20 per cent
DAs are more massive than 0.8M� in our bright and hot sample,
of the same order as that discovered by Liebert et al. (2005) for
the PG sample. The DB distribution is interesting, however, as it
tends to significantly higher masses than does the DA distribution!
We found only two stars from our sample with published atmo-
spheric parameters in the literature, with �Teff = 510 ± 30 and
� log g = 0.12 ± 0.15, so we could not do a comparison as
Kleinman et al. (2004) and Eisenstein et al. (2006) did for the DA
results.
7 E X T R E M E M A S S S TA R S
Nalezyty & Madej (2004) published a catalogue of all massive white
dwarf stars then known, with 112 stars more massive than 0.8M�.
The four stars with M � 1.3M� in their list are magnetic ones
and therefore have large uncertainties in their estimated masses.
Dahn et al. (2004) found one non-magnetic massive white dwarf,
LHS 4033, with M � 1.318–1.335M�, depending on the core
composition. Our oxygen–neon core mass for their derived Teff =10 900 K and log g = 9.46 is M � 1.30M�. We note that the
models from the Montreal group used to derive Teff and log g in Dahn
et al. (2004) show the same increase in mass with decreasing Teff
as our models and therefore we do not take this mass determination
as completely reliable due to the objects relatively low temperature.
For GD 50 (WD J0348−0058), Dobbie et al. (2006) found Teff =41 550 ± 720 K and log g = 9.15 ± 0.05. Our oxygen–neon core
mass for their derived Teff and log g is M � 1.23±0.02M�, very
similar to the value reported by them for C/O models. They also
show this massive star is consistent with its formation and evolution
as a single star, not the product of a merger.
From the 7755 pure DA white dwarf stars, we found 1611 (22 per
cent) withM > 0.8M�. For the 2945 stars brighter than g = 19 we
found 760 (26 per cent) with M > 0.8M�, but for the 1733 stars
brighter than g = 19 and hotter than Teff = 12 000 K, we find only
105 stars (6 per cent) with M > 0.8M�. The most massive star in
our hot and bright sample is SDSS J075916.53+433518.9, whose
spectrum spSpec-51883-0436-045 is shown in Fig. 10, with g =18.73, Teff = 22100 ± 450 K, log g = 9.62 ± 0.07, M = 1.33 ±0.01M� and estimated distance of d = 104 ± 4 pc. We caution that
4000 5000 6000 7000
0
0.5
1
1.5
2
log g=9
log g=10
Figure 10. Spectrum of SDSS J075916.53+433518.9 with g = 18.73,
Teff = 22 100 K and two models, with log g = 9 and 10. The higher log gfits the Hα line better, but the lower log g fits the higher lines better, where
the SNR is smaller. This low SNR is typical for the stars closer to our upper
cut-off of g = 19.
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1322 S. O. Kepler et al.
Table 6. DA stars with masses above 1.2M� and below 1.3M� derived from the SDSS spectra, with Teff � 12 000 K.
Spectra-M-P-F Name g Mg Teff σT log g σg M σM d(K) (K) (M�) (M�) (pc)
spSpec-51691-0342-639 SDSS J155238.21+003910.3 18.44 12.23 15 924 387 9.280 0.050 1.262 0.010 97
spSpec-51915-0453-540 SDSS J094655.94+600623.4 17.99 10.87 28 125 220 9.370 0.040 1.287 0.010 123
spSpec-52374-0853-198 SDSS J133420.97+041751.1 18.52 12.34 17 549 422 9.150 0.060 1.223 0.020 125
spSpec-52703-1165-306 SDSS J150409.88+513729.1 18.84 10.28 79 873 8228 9.050 0.390 1.204 0.100 468
spSpec-52751-1221-177 SDSS J110735.32+085924.5 18.42 12.23 18 715 327 9.140 0.060 1.219 0.020 128
spSpec-52872-1402-145 SDSS J154305.67+343223.6 18.33 10.85 30 472 313 9.300 0.070 1.269 0.010 168
4000 5000 6000 7000
0
1
2
log g=9.0
log g=9.5
Figure 11. Spectrum of SDSS J094655.94+600623.4 with g = 17.99,
Teff = 28 100 K and two models, with log g = 9 and 9.5. This spectrum
is typical of the SNR achieved for the 1003 DAs and 59 DBs brighter than
g = 18 in our sample.
the evolutionary models used to estimate the radii, and therefore the
masses, in our analysis do not include post-Newtonian corrections,
important for masses above M � 1.30M� (Chandrasekhar &
Tooper 1964). For the stars brighter than g = 19, we find 21 others
with masses larger thanM = 1.3M�, all below Teff = 9000 K. We
deem the mass determinations for stars cooler than Teff � 12 000 K
unreliable. In Table 6, we list the DAs with 1.2 < M < 1.3M�hotter than Teff = 12 000 K.
The spectrum for the brighter g = 17.99 SDSS
J094655.94+600623.4 is shown in Fig. 11. Because our analysis
uses relatively low SNR spectra and gravity effects dominate
mainly below 3800 Å, where we have no data, our conclusion is
that we must undertake a study in the violet or ultraviolet to measure
the masses more accurately. An extensive study of gravitational
redshift would also be critical.
For the 507 single DBs we find 30 DBs with log g > 9. Most of
our massive DBs are cooler than Teff � 16 000 K, or fainter than
g = 19, except for SDSS J213103.39+105956.1 with g = 18.80,
Teff = 16 476 ± 382 and log g = 9.64 ± 0.21, corresponding to a
Figure 12. Evolutionary tracks for He white dwarf stars calculated by
Althaus et al. (2001) and the location of the lowest mass stars in our sample.
Most of these stars are below g = 19 and therefore have noisy spectra.
massM = 1.33±0.04M�, and for SDSS J224027.11−005945.5
with g = 18.82, Teff = 17 260 ± 402, and log g = 9.31 ± 0.20,
corresponding to a mass M = 1.25 ± 0.06M�.
The low-mass stars present in our sample are consistent with
He core evolution models calculated by Althaus et al. (2001), and
displayed in Fig. 12. It is important to stress that these stars should
be studied with more accurate spectra and model atmospheres, as
they are possible progenitors of Type Ia supernova if they accrete
mass from companions.
8 C O N C L U S I O N S
Our investigations into the mass distribution of the SDSS DR4 white
dwarf sample from Eisenstein et al. (2006) revealed several items.
First, all groups are seeing nearly identical increases in mean DA at
lower temperatures (less than ≈12 000 K for DAs and 16 000 K for
DBs). Either this is truly going on in the white dwarf stars, or there
is missing or incorrect physics in everyone’s models. We propose
the treatment of neutral particles as the most likely explanation. We
suspect the atmospheric models should be improved with a detailed
inclusion of the line broadening by neutral particles, since the in-
crease in apparent mass for both DAs and DBs occur at temperatures
when recombination becomes important.
Secondly, we find a significant difference between the DA and
DB mass distributions, with the DB distribution significantly more
weighted to massive stars. Figs 13 and 14 show the combined DA
and DB histograms. The DB histograms have been renormalized
to the DA maximum for display purposes. Our results contradict
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White dwarf mass distribution in the SDSS 1323
0.2 0.4 0.6 0.8 1 1.2 1.4
0
100
200
300
Sloan DR4
DB
DA
Figure 13. DA and DB histograms for comparison. The DB histogram has
been renormalized to the DA maximum for display purposes.
0.2 0.4 0.6 0.8 1 1.2 1.4
0
SDSS DR4
DBDA
Figure 14. DA and DB histograms corrected by observed volume for com-
parison. The DB histogram has been renormalized to the DA maximum for
display.
nearly all previous work which show the mean DA and DB masses
to be similar (with the exception of Koester et al. 2001). We note that
the previous efforts, though, were based on histograms for DBs with
less than 50 stars, and our DB histogram has 150 stars. However, we
still need to explore our DB models and fits in more detail to verify
the validity of this novel result. Specifically, we find 〈M〉DB =0.711 ± 0.009M�, higher than 〈M〉DA = 0.593 ± 0.016M� for
the 1733 DAs brighter than g = 19, and hotter than Teff = 12 000 K.
This is a significant new result and must be investigated further.
We have also detected a large number of massive DA white dwarf
stars: 760 with M > 0.8M� brighter than g = 19 and 105 both
brighter than g = 19 and hotter than Teff = 12 000 K. We report the
highest log g white dwarf ever detected.
AC K N OW L E D G M E N T S
We thank Dr Daniel Eisenstein for some helpful suggestions, and his
AUTOFIT code, and the referee, Dr Martin Barstow, for very detailed
and useful comments that improved the presentation of the paper.
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